It's a great idea but is it legal to operate a personal Stingray?

Yes! Well let me be a bit more clear. The FCC has a process for, and indicated they would likely approve, a special temporary operating license for such a device given that it would only operate for a short time (during the break-in) and not interfere with the thieve's or neighbor's cell phones as it would respond with no compatible roaming network. Also it wouldn't do any sort of interception or monitoring other than the tower ping.

It would also need permission from the licensed spectrum holder, that could be harder to get.

But yes, completely legit with the right paperwork.

Wow, I'm a bit surprised by that. Thank you for answering.

I was a little as well, I happened to be at an SDR conference and one of the attendees was a lawyer working with the FCC on licensing issues[1]. I gave him my use case and he said that if it was constrained the way I said it would be, then if it came through their office they would likely approve it.

[1] The FCC has been pretty engaged in understanding the impacts of SDRs because of their potential for abuse (as one attendee called it, "The Drones of spectrum." As an amateur (AI6ZR) and someone in industry I've been helping as an SME to explain some of this stuff to their less technical staff.

And investigators probably would not care — consider how many identity theft cases have the ID thief on camera (in my case, at a hardware store opening up a line of credit in sight of store cameras, and later using a cloned card at an ATM machine, surely in view of an ATM camera.)...yet despite thousands stolen in ID theft cases, there is hardly ever any action taken.

My thought would be they could track the phone of the person, who would probably still have it on them, and using that and a fuzzy security camera image have enough evidence to arrest them at least.